We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n-2)-connected. In addition, we show that for every non-trivial free factor system of a free group, the corresponding relative free splitting complex is contractible.

中文翻译：

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自由群自由因子复合体的同伦型

我们表明，等级为n> 1的自由组的自由因子的复数是同构的，等效于尺寸为n-2的球体的楔形。我们还证明，当n> 1时，自由分裂复合体中（未缩减的）外层空间的补体是同质的，等同于自由因子系统的复合体，而且是（n-2）连通的。此外，我们表明，对于一个自由组的每个非平凡的自由因子系统，相应的相对自由分裂复合体是可收缩的。